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Lectures on measures on limit sets of Kleinian groups. (English) Zbl 0611.30036
Analytical and geometric aspects of hyperbolic space, Symp. Warwick and Durham/Engl. 1984, Lond. Math. Soc. Lect. Note Ser. 111, 281-323 (1987).
[For the entire collection see Zbl 0601.00008.]
This paper contains an expanded version of five lectures delivered by the author. It is a beautiful exposition of topics concerning the analytic properties of the limit sets of Kleinian groups in n dimensions. The first lecture is introductory and defines the classes of problems discussed. The exponent of convergence of Poincaré series is defined. In the second lecture, the Patterson-Sullvian measures are defined and their basic properties are derived. The third lecture focuses on the distribution of orbits and contains various relationships between the exponent of convergence for a group, Hausdorff measure and other properties of the limit sets of Kleinian groups. In the last two lectures, the previous concepts are related to the spectrum of the Laplacian and to geodesic flows.
Reviewer: W.Abikoff

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)